Factorable Monoids

Abstract

We study groups and monoids that are equipped with an extra structure called factorability.<br /> A factorable group can be thought of as a group G together with the choice of a generating set S and a particularly well-behaved normal form map G → S^*, where S^* denotes the free group over S. This is related to the theory of complete rewriting systems, collapsing schemes and discrete Morse theory.<br /> Given a factorable monoid M, we construct new resolutions of <b>Z</b> over the monoid ring <b>Z</b>M. These resolutions are often considerably smaller than the bar resolution E_*M. <br /> As an example, we show that a large class of generalized Thompson groups and monoids fits into the framework of factorability and compute their homology groups. In particular, we provide a purely combinatorial way of computing the homology of Thompson's group F.